[Salb] , “On the density of rational and integral points on algebraic varieties”, to appear in J.. The closed geodesics on the modular surface which are equiva-lent to themselves when t
Trang 1j ≤ 4 that C8−3/2 √ d
1 (C2C3C4)4 ≤ (C1 C2C3C4)5−3/8 √ d and C8−g(d)
1 (C2C3C4)4 ≤
(C1C2C3C4)5−g(d)/4 Hence if we sum over all dyadic intervals [C j , 2C j ], 1 ≤ j ≤ 4
for 2-powers C j as above and argue as in (op cit.), we will get a set of d,ε
B 15(3/2 √
d)/16+5/4+ε rational lines on X such that there are O d,ε (B 15g(d)/16+5/4+ε) rational points of height≤ B on the union of all geometrically integral hyperplane
sections Π∩ X with H(Π) ≤ (5B) 1/4 which do not lie on these lines
We now combine this with (1.1) and (2.3) Then we conclude that there are
d,ε B 15g(d)/16+5/4+ε +B 5/2+ε points in S(X, B) outside these lines if X is not a
cone of a Steiner surface and ε B 15g(d)/16+5/4+ε +B 11/4+ε such points if X is a cone of a Steiner surface To finish the proof, note that max (15g(d)/16+5/4, 5/2) = max (45/16 √
d + 5/4, 5/2) in the first case and that max (15g(d)/16 + 5/4, 11/4) =
4 The points on the lines
We shall in this section estimate the contribution to N (X , B) from the lines
in Theorem 3.3
Lemma 4.1 X ⊂ P4 be a geometrically integral projective threefold over Q of
degree d ≥ 2 Let M be a set of O d,ε (B 5/2+ε ) rational lines Λ on X each contained
in some hyperplane Π ⊂ P4 of height ≤ (5B) 1/4 and N ( ∪Λ ∈M Λ, B) be the number
of points in S(X, B) ∩ (∪Λ ∈MΛ (Q)) Then the following holds.
(a) N ( ∪Λ ∈M Λ, B) = O d,ε (B 11/4+ε + B 5/2+3/2d+ε ).
(b) N ( ∪Λ ∈M Λ, B) = O d,ε (B 5/2+3/2d+ε ) if X is not a cone over a curve.
(c) N ( ∪Λ ∈M Λ, B) = O d,ε (B 5/2+ε + B 9/4+3/2d+ε ) if there are only finitely
many planes on X.
Proof We shall for each Λ∈ M choose a hyperplane Π(Λ) ⊂ P4 of minimal
height containing Λ Then, H(Π(Λ)) ≤ κH(Λ) 1/3 for some absolute constant κ
by (1.2) The contribution to N ( ∪Λ ∈M Λ, B) from all Λ ∈ M where Π(Λ) ∩ X
is not geometrically integral is O d,ε (B 11/4+ε ) in general and O d,ε (B 5/2+ε ) if X is
not a cone over a curve (see (2.3)) The contribution from the lines Λ∈ M with
N (Λ, B) ≤ 1 is O d,ε (B 5/2+ε) We may and shall therefore in the sequel assume that Π(Λ)∩ X is geometrically integral and N(Λ, B) ≥ 2 for all Λ ∈ M From
N (Λ, B) ≥ 2 we deduce that H(Λ) ≤ 2B2, N (Λ, B) B2/H(Λ) and
Λ∈M
N (Λ, B) B2
Λ∈M
H(Λ) −1
It is therefore sufficient to prove that:
Λ∈M
H(Λ) −1 = O
d,ε (B 1/2+3/2d+ε)
in general and that
Λ∈M
H(Λ) −1 = O
d,ε (B 1/2+ε + B 1/4+3/2d+ε)
if there are only finitely many planes on X.
A proof of (4.3) may be found in the proofs of Lemma 3.2.2 and Lemma 3.2.3
in [BS04] It is therefore enough to show (4.4).
Trang 2Let M1 ⊆ M be the subset of rational lines Λ such that Π(Λ) ∩ X contains
only finitely many lines Then it is known and easy to show using Hilbert schemes
that there is a uniform upper bound depending only on d for the number of lines
on such Π(Λ)∩ X There can therefore only be O d (B 5/4) lines Λ ∈ M1 as there
are only O(B 5/4) possibilities for Π(Λ) The contribution to
Λ∈M1H(Λ) −1 from
lines of height ≥ B 3/4 is thus O d (B 1/2)
Now let [R, 2R] be a dyadic interval with 1 ≤ R ≤ B 3/4 Then H(Π(Λ)) H(Λ) 1/3 R 1/3 for Λ with H(Λ) ∈ [R, 2R] so that there are only O(R 5/3)
possi-bilities for Π(Λ) and O d (R 5/3) such lines Λ The contribution to
Λ∈M1H(Λ) −1 from the lines of height H(Λ) ∈ [R, 2R] is thus O d (R 2/3) Hence if we sum over all
O(log B) dyadic intervals [R, 2R] with R ≤ B 3/4 we get that lines of height≤ B 3/4
contribute with O d (B 1/2 (log B)) to
Λ∈M1
H(Λ) −1 so that
Λ∈M1
H(Λ) −1 = O
d (B 1/2 (log B)).
We now consider the subset M2⊆ M of rational lines Λ where Π(Λ) ∩ X contains
infinitely many lines This is equivalent to (Π(Λ)∩ X)( ¯Q) being a union of its lines
(see [Salb], 7.4).
There exists therefore by [Salb], 7.8 a hypersurface W ⊂ P4∨ of degree O
d(1) such that any hyperplane Π ⊂ P4 where Π∩ X contains infinitely many lines is
parameterised by a point on W There are thus O d (B) such hyperplanes of height
≤ (5B) 1/4 If R ≥ 1, then H(Π(Λ)) H(Λ) 1/3 R 4/3 for lines Λ of height
≤ 2R There are therefore O d (R 4/3) possibilities for Π(Λ) among all lines of height
≤ 2R There are also by the proof of lemma 3.2.2 in [BS04] O d,ε (R 2/d+ε) rational lines of height ≤ 2R on each geometrically integral hyperplane section There are
thus d,ε min(BR 2/d+ε , R 4/3+2/d+ε) rational lines of height ≤ 2R in M2 Hence
if R ≥ 1, the contribution from all rational lines of height H(Λ) ∈ [R, 2R] to
Λ∈M2H(Λ) −1 will be d,ε min(BR −1+2/d+ε , R 1/3+2/d+ε)≤ B 1/4+3/2d R ε If we
cover [1, 2B2] by O(log B) dyadic intervals with 1 ≤ R ≤ B2 , we obtain
Λ∈M2
H(Λ) −1 = O
d (B 1/4+3/2d+ε )
If we combine (4.5) and (4.6), then we get (4.4) This completes the proof
5 Proof of the theorems
We shall in this section prove Theorems 0.1 and 0.2
Theorem 5.1 Let X ⊂ P4 be a geometrically integral projective hypersurface over Q of degree d ≥ 3 Then,
N (X, B) = O d,ε (B 11/4+ε + B 3/2d+5/2+ε )
If X is not a cone over a curve, then,
N (X, B) = O d,ε (B 45/16 √
d+5/4+ε + B 3/2d+5/2+ε )
Trang 3If there are only finitely many planes on X ⊂ P4, then
N (X, B) =
O ε (B 45/16 √
d+5/4+ε) if d = 3 or 4 and X is not a cone of a
Steiner surface
O ε (B 1205/448+ε) if d = 4
O ε (B 51/20+ε) if d = 5
O d,ε (B 5/2+ε) if d ≥ 6.
Proof Let h(d) = max(45/16 √
d + 5/4, 5/2) if X is not a cone of a Steiner
surface and h(d) = 1205/448 if X is a cone of a Steiner surface Then it is shown in Theorem 3.3 that there is a set M of O d,ε (B 45/32 √
d+5/4+ε ) rational lines on X such that all but O d,ε (B h(d)+ε) points in S(X, B) lie on the union of these lines To count
the points in S(X, B) ∩ (∪Λ ∈MΛ (Q)), we note that #M = O d,ε (B 5/2+ε) and apply Lemma 4.1 We then get that N (X, B) = O d,ε (B h(d)+ε + B 11/4+ε + B 3/2d+5/2+ε) in
general Further, if X is not a cone over a curve, then N (X, B) = O d,ε (B h(d)+ε+
B 3/2d+5/2+ε) while N (X, B) = O d,ε (B h(d)+ε + B 3/2d+9/4+ε + B 5/2+ε) in the more
special case when there are only finitely many planes on X ⊂ P4 It is now easy to complete the proof by comparing all the exponents that occur Corollary 5.2 Let X ⊂ P4be a geometrically integral projective hypersur-face over Q of degree d ≥ 3 Then, N(X, B) = O d,ε (B 3+ε)
Proof This follows from the first assertion in Theorem 5.1 and [BS04],
Lemma 3.1.1 (The result was first proved for d ≥ 4 in [BS04] and then for
Theorem 5.3 Let X ⊂ P n be a geometrically integral projective threefold over
Q of degree d Let X be the complement of the union of all planes on X Then,
N (X , B) =
O n,ε (B15√
3/16+5/4+ε) if d = 3,
O n,ε (B 1205/448+ε) if d = 4,
O n,ε (B 51/20+ε) if d = 5,
O d,n,ε (B 5/2+ε) if d ≥ 6
If n = d = 4 and X is not a cone of a Steiner surface, then
N (X , B) = O
n,ε (B 85/32+ε )
Proof If n = 4 and there are only finitely many planes on X ⊂ P4, then this
follows from Theorem 5.1 since N (X , B) ≤ N (X, B) If there are infinitely many
planes on X, then X is empty [Salb], 7.4 and N (X , B) = 0 To prove Theorem 5.3 for n > 4, we reduce to the case n = 4 by means of a birational projection
Proposition 5.4 Let k be an algebraically closed field of characteristic 0 and (a0, , a5), (b0, , b5) be two sextuples in k ∗ = k \ {0} and X ⊂ P5 be the closed subscheme defined by the two equations a0x e + .+a5x e = 0 and b0x f0+ .+b5x f5 =
0 where e < f Then the following holds
(a) There are only finitely many singular points on X,
(b) Xis a normal integral scheme of degree ef ,
(c) There are only finitely many planes on X if f ≥ 3,
(d) X is not a cone over a Steiner surface.
Trang 4Proof (a) Let (x0, , x5) be a singular point on X with at least two non-zero
coordinates x i , x j Then it follows from the Jacobian criterion that a i b j x e −1
i x f −1
a j b i x e −1
j x f −1
i and hence that a i b j x f −e
j = a j b i x f −e
i Hence there are only f − e
possible values for x j /x i for any two non-zero coordinates x i , x jof a singular point
This implies that there only finitely many singular points on X.
(b) The forms a0xe + + a5xe and b0xf0 + + b5xf5 are irreducible for
(a0, , a5), (b0, , b5) as above and define integral hypersurfaces Y a ⊂ P5 and
Y b ⊂ P5 of different degrees Therefore, X ⊂ P5 is a complete intersection of
codimension two of degree ef In particular, Y b is a Cohen-Macaulay scheme and
X ⊂ Y b a closed subscheme which is regularly immersed Hence, as the singular
locus of X is of codimension ≥ 2, we obtain from [AK70], VII 2.14, that X is
normal As X is of finite type over k, it is thus integral if and only if it is connected.
To show that X is connected, use Exercise II.8.4 in [Har77].
(c) It is known that there are only finitely many planes on non-singular hyper-surfaces of degree≥3 in P5 (see [Sta06] where it is attributed to Debarre) There
are thus only finitely many planes on Y and hence also on X.
(d) It is well known that a Steiner surface has three double lines The singular
locus of a cone of Steiner surface is thus two-dimensional Hence X cannot be such
Parts (a) and (c) of the previous proposition were used already in [Kon02] and [BHB].
Theorem 5.5 Let (a0, , a5) and (b0, , b5) be two sextuples of rational
numbers different from zero and e < f be positive integers with f ≥ 3 Let X ⊂ P5
be the threefold defined by the two equations a0x e + + a5xe = 0 and b0xf0 +
+ b5x f5 = 0 Then there are only finitely many planes on X If X ⊂ X is the complement of these planes in X, then
N (X , B) = O
ε (B 45/16 √ ef +5/4+ε
) if ef = 3 or 4,
N (X , B) = O ε (B 51/20+ε) if ef = 5,
N (X , B) = O e,f,ε (B 5/2+ε) if ef ≥ 6
References
[AK70] A Altman& S Kleiman – Introduction to Grothendieck duality theory, Lecture Notes
in Mathematics, Vol 146, Springer-Verlag, Berlin, 1970.
[BHB] T D Browning & D R Heath-Brown – “Simultaneous equal sums of three powers”,
Proc of the session on Diophantine geometry (Pisa, 1st April-30th July , 2005), to appear.
[BHB05] , “Counting rational points on hypersurfaces”, J Reine Angew Math 584
(2005), p 83–115.
[BS04] N Broberg& P Salberger – “Counting rational points on threefolds”, in Arithmetic
of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr Math., vol 226,
Birkh¨ auser Boston, Boston, MA, 2004, p 105–120.
[Gre97] G Greaves– “Some Diophantine equations with almost all solutions trivial”,
Mathe-matika 44 (1997), no 1, p 14–36.
[Har77] R Hartshorne– Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts
in Mathematics, No 52.
[HB02] D R Heath-Brown– “The density of rational points on curves and surfaces”, Ann.
of Math (2) 155 (2002), no 2, p 553–595.
Trang 5[Kon02] A Kontogeorgis – “Automorphisms of Fermat-like varieties”, Manuscripta Math 107
(2002), no 2, p 187–205.
[Rog94] E Rogora– “Varieties with many lines”, Manuscripta Math 82 (1994), no 2, p 207–
226.
[Sala] P Salberger – “Counting rational points on projective varieties”, preprint.
[Salb] , “On the density of rational and integral points on algebraic varieties”, to appear
in J Reine und Angew Math., (see www.arxiv.org 2005).
[Salc] , “Rational points of bounded height on projective surfaces”, submitted.
[Sal05] P Salberger– “Counting rational points on hypersurfaces of low dimension”, Ann.
Sci ´ Ecole Norm Sup (4) 38 (2005), no 1, p 93–115.
[Sch91] W M Schmidt – Diophantine approximations and Diophantine equations, Lecture
Notes in Mathematics, vol 1467, Springer-Verlag, Berlin, 1991.
[Sha99] I R Shafarevich(ed.) – Algebraic geometry V, Encyclopaedia of Mathematical
Sci-ences, vol 47, Springer-Verlag, Berlin, 1999, Fano varieties, A translation of Algebraic
geometry 5 (Russian), Ross Akad Nauk, Vseross Inst Nauchn i Tekhn Inform.,
Moscow, Translation edited by A N Parshin and I R Shafarevich.
[SR49] J G Semple& L Roth – Introduction to Algebraic Geometry, Oxford, at the
Claren-don Press, 1949.
[Sta06] J M Starr– “Appendix to the paper: The density of rational points on non-singular
hypersurfaces, II by T.D Browning and R Heath-Brown”, Proc London Math Soc.
93 (2006), p 273–303.
[SW97] C M Skinner & T D Wooley – “On the paucity of non-diagonal solutions in certain
diagonal Diophantine systems”, Quart J Math Oxford Ser (2) 48 (1997), no 190,
p 255–277.
[TW99] W Y Tsui & T D Wooley – “The paucity problem for simultaneous quadratic and
biquadratic equations”, Math Proc Cambridge Philos Soc 126 (1999), no 2, p 209–
221.
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number theory, Vol 2 (Allerton Park, IL, 1995), Progr Math., vol 139, Birkh¨auser Boston, Boston, MA, 1996, p 803–815.
Department of Mathematics, Chalmers University of Technology, SE-41296 G¨ oteborg,
Sweden
E-mail address: salberg@math.chalmers.se
Trang 6Volume 7, 2007
Reciprocal Geodesics
Peter Sarnak
Abstract The closed geodesics on the modular surface which are
equiva-lent to themselves when their orientation is reversed have recently arisen in
a number of different contexts.We examine their relation to Gauss’
ambigu-ous binary quadratic forms and to elements of order four in his composition
groups.We give a parametrization of these geodesics and use this to count them
asymptotically and to investigate their distribution.
This note is concerned with parametrizing, counting and equidistribution of
conjugacy classes of infinite maximal dihedral subgroups of Γ = P SL(2,Z) and their connection to Gauss’ ambiguous quadratic forms These subgroups feature in the recent work of Connolly and Davis on invariants for the connect sum problem
for manifolds [CD] They also come up in [PR04] (also see the references therein)
in connection with the stability of kicked dynamics of torus automorphisms as well
as in the theory of quasimorphisms of Γ In [GS80] they arise when classifying
codimension one foliations of torus bundles over the circle Apparently they are of quite wide interest As pointed out to me by Peter Doyle, these conjugacy classes and the corresponding reciprocal geodesics, are already discussed in a couple of
places in the volumes of Fricke and Klein ([FK], Vol I, page 269, Vol II, page
165) The discussion below essentially reproduces a (long) letter that I wrote to Jim Davis (June, 2005)
Denote by {γ}Γ the conjugacy class in Γ of an element γ ∈ Γ The elliptic
and parabolic classes (i.e., those with t(γ) ≤ 2 where t(γ) = |trace γ|) are
well-known through examining the standard fundamental domain for Γ as it acts on
H We restrict our attention to hyperbolic γ’s and we call such a γ primitive (or prime) if it is not a proper power of another element of Γ Denote by P the set of
such elements and by Π the corresponding set of conjugacy classes The primitive
elements generate the maximal hyperbolic cyclic subgroups of Γ We call a p ∈ P
reciprocal if p −1 = S −1 pS for some S ∈ Γ In this case, S2= 1 (proofs of this and
further claims are given below) and S is unique up to multiplication on the left by
γ ∈ p Let R denote the set of such reciprocal elements For r ∈ R the group
D r =r, S, depends only on r and it is a maximal infinite dihedral subgroup of
2000 Mathematics Subject Classification Primary 11F06, Secondary 11M36.
Key words and phrases Number theory, binary quadratic forms, modular surface.
Supported in part by the NSF Grant No DMS0500191 and a Veblen Grant from the IAS.
c
2007 Peter Sarnak
Trang 7Γ Moreover, all of the latter arise in this way Thus, the determination of the conjugacy classes of these dihedral subgroups is the same as determining ρ, the subset of Π consisting of conjugacy classes of reciprocal elements Geometrically,
each p ∈ P gives rise to an oriented primitive closed geodesic on Γ\H, whose length
is log N (p) where N (p) =
t(p) +
t(p)2− 4/2
2
Conjugate elements give rise to the same oriented closed geodesic A closed geodesic is equivalent to itself with its orientation reversed iff it corresponds to an{r} ∈ρ
The question as to whether a given γ is conjugate to γ −1 in Γ is reflected in part in the corresponding local question If p ≡ 3 (mod 4), then c =
is not conjugate to c −1 in SL(2,Fp ), on the other hand, if p ≡ 1 (mod 4) then
every c ∈ SL(2, F p ) is conjugate to c −1 This difficulty of being conjugate in G( ¯ F ) but not in G(F ) does not arise if G = GL n (F a field) and it is the source of a basic general difficulty associated with conjugacy classes in G and the (adelic) trace
formula and its stabilization [Lan79] For the case at hand when working overZ, there is the added issue associated with the lack of a local to global principle and
in particular the class group enters In fact, certain elements of order dividing four
in Gauss’ composition group play a critical role in the analysis of the reciprocal classes
In order to study ρ it is convenient to introduce some other set theoretic involutions of Π Let φ R be the involution of Γ given by φ R (γ) = γ −1. Let
φ w (γ) = w −1 γw where w =
∈ P GL(2, Z) (modulo inner
automor-phism φ w generates the outer automorphisms of Γ coming from P GL(2, Z)) φ R and φ w commute and set φ A = φ R ◦φ w = φ w ◦φ R These three involutions generate
the Klein group G of order 4 The action of G on Γ preserves P and Π For H
a subgroup of G, let Π H ={{p} ∈ Π : φ({p}) = {p} for φ ∈ H} Thus Π {e} = Π and Πφ R =ρ We call the elements in Πφ A ambiguous classes (we will see that
they are related to Gauss’ ambiguous classes of quadratic forms) and of Πφ w , inert classes Note that the involution γ → γ t is, up to conjugacy in Γ, the same as φ R, since the contragredient satisfies t g −1 =
−1 0
g
−1 0
Thus p ∈ P is
reciprocal iff p is conjugate to p t
To give an explicit parametrization ofρlet
(1) C = (a, b) ∈ Z2: (a, b) = 1, a > 0, d = 4a2+ b2 is not a square
.
To each (a, b) ∈ C let (t0, u0 ) be the least solution with t0 > 0 and u0 > 0 of
the Pell equation
Define ψ : C −→ρby
t0 − bu0
au0 t0+ bu0
2
Γ
,
Trang 8It is clear that ψ((a, b)) is reciprocal since an A ∈ Γ is symmetric iff S0−1 AS0= A −1 where S0 =
−1 0
Our central assertion concerning parametrizingρis; Proposition 1 ψ : C −→ρ is two-to-one and onto ∗
There is a further stratification to the correspondence (3) Let
(4) D = {m | m > 0 , m ≡ 0, 1 (mod 4) , m not a square}
Then
d ∈D
C d
where
.
Elementary considerations concerning proper representations of integers as a sum
of two squares shows that C d is empty unless d has only prime divisors p with p ≡ 1
(mod 4) or the prime 2 which can occur to exponent α = 0, 2 or 3 Denote this
subset ofD by D R Moreover for d ∈ D R,
where for any d ∈ D, ν(d) is the number of genera of binary quadratic forms of
discriminant d ((6) is not a coincidence as will be explained below) Explicitly ν(d)
is given as follows: If d = 2 α D with D odd and if λ is the number of distinct prime
divisors of D then
2λ −1 if α = 0
2λ −1 if α = 2 and D ≡ 1 (mod 4)
2λ if α = 2 and D ≡ 3 (mod 4)
2λ if α = 3 or 4
2λ+1 if α ≥ 5
Corresponding to (5) we have
d ∈D R
ρd ,
withρd = ψ(C d ) In particular, ψ : C d −→ρd is two-to-one and onto and hence
Local considerations show that for d ∈ D the Pell equation
can only have a solution if d ∈ D R When d ∈ D Rit may or may not have a solution Let D −
R be those d’s for which (9) has a solution and D+
R the set of d ∈ D R for which (9) has no integer solution Then
(i) For d ∈ D+
R none of the{r} ∈ρd, are ambiguous
(ii) For d ∈ D −
R, every{r} ∈ρd is ambiguous
∗Part of this Proposition is noted in ([FK], Vol I, pages 267-269).
Trang 9In this last case (ii) we can choose an explicit section of the two-to-one map
(3) For d ∈ D −
R let C −
d ={(a, b) : b < 0}, then ψ : C −
d −→ ρd is a bijection.†
Using these parameterizations as well as some standard techniques from the spectral theory of Γ\H one can count the number of primitive reciprocal classes.
We order the primes{p} ∈ Π by their trace t(p) (this is equivalent to ordering the
corresponding prime geodesics by their lengths) For H a subgroup of G and x > 2
let
{p} ∈ΠH t(p) ≤ x
1
Theorem 2 As x −→ ∞ we have the following asymptotics:
2 log x ,
8π2x(log x)2,
8x ,
2 log x
and
8π x
1/2
log x
(All of these are established with an exponent saving for the remainder).
In particular, roughly the square root of all the primitive classes are reciprocal while the fourth root of them are simultaneously reciprocal ambiguous and inert
We turn to the proofs of the above statements as well as a further discussion connectingρwith elements of order dividing four in Gauss’ composition groups
We begin with the implication S −1 pS = p −1=⇒ S2= 1 This is true already
in P SL(2, R) Indeed, in this group p is conjugate to ±
λ 0
0 λ −1
with λ > 1.
Hence Sp −1 = pS with S =
a b
c d
=⇒ a = d = 0, i.e., S = ±
−β −1 0
and so S2 = 1 If S and S1 satisfy x −1 px = p −1 then SS −1
1 ∈ Γ p the centralizer
of p in Γ But Γ p = p and hence S = βS1 with β ∈ p Now every element
S ∈ Γ whose order is two (i.e., an elliptic element of order 2) is conjugate in Γ to
S0=±
−1 0
Hence any r ∈ R is conjugate to an element γ ∈ Γ for which
S −1
0 γS0 = γ −1 The last is equivalent to γ being symmetric Thus each r ∈ R is
We can be more precise:
Lemma 3 Every r ∈ R is conjugate to exactly four γ’s which are symmetric.
† For a general d ∈ D+ it appears to be difficult to determine explicitly a one-to-one section
of ψ.
Trang 10To see this associate to each S satisfying
the two solutions γ S and γ
S (here γ
S = Sγ S) of
Then
S rγ S = ((γ
S)−1 rγ
S)−1 and both of these are symmetric.
Thus each S satisfying (17) affords a conjugation of r to a pair of inverse symmetric matrices Conversely every such conjugation of r to a symmetric matrix is induced
as above from a γ S Indeed if β −1 rβ is symmetric then S −1
0 β −1 rβS
0= β −1 r −1 β and so βS −1
0 β −1 = S for an S satisfying (17) Thus to establish (16) it remains to count the number of distinct images γ −1
S rγ S and its inverse that we get as we vary
over all S satisfying (17) Suppose then that
S rγ S = γ −1
S rγ S
Then
S = b ∈ Γ r = r
Also from (18)
S Sγ S = γ −1
S S γ
S
or
S S γ S γ −1
S = S .
Using (21) in (23) yields
But bS satisfies (17), hence bSbS = 1 Putting this relation in (24) yields
These steps after (22) may all be reversed and we find that (20) holds iff S = b2S for some b ∈ Γ r Since the solutions of (17) are parametrized by bS with b ∈ Γ r(and
S a fixed solution) it follows that as S runs over solutions of (17), γ −1
S rγ S and
(γ
S)−1 r(γ
S) run over exactly four elements This completes the proof of (16) This
argument should be compared with the one in ([Cas82], p 342) for counting the
number of ambiguous classes of forms Peter Doyle notes that the four primitive symmetric elements which are related by conjugacy can be described as follows: If
A is positive, one can write A as γ γ with γ ∈ Γ (the map γ −→ γ γ is onto such); then A, A −1 , B, B −1 , with B = γγ , are the four such elements.
To continue we make use of the explicit correspondence between Π and classes
of binary quadratic forms (see [Sar] and also ([Hej83], pp 514-518). ‡ An integral binary quadratic form f = [a, b, c] (i.e ax2+ bxy + xy2) is primitive if (a, b, c) = 1 Let F denote the set of such forms whose discriminant d = b2− 4ac is in D Thus
d ∈D
F d
with F d consisting of the forms of discriminant d The symmetric square represen-tation of P GL2gives an action σ(γ) on F for each γ ∈ Γ It is given by σ(γ)f = f
‡This seems to have been first observed in ([FK], Vol., page 268)